A hybrid immersed boundary and material point method for simulating 3D fluid–structure interaction problems

A numerical method is developed for solving the 3D, unsteady, incompressible Navier–Stokes equations in curvilinear coordinates containing immersed boundaries (IBs) of arbitrary geometrical complexity moving and deforming under forces acting on the body. Since simulations of flow in complex geometries with deformable surfaces require special treatment, the present approach combines a hybrid immersed boundary method (HIBM) for handling complex moving boundaries and a material point method (MPM) for resolving structural stresses and movement. This combined HIBM & MPM approach is presented as an effective approach for solving fluid–structure interaction (FSI) problems. In the HIBM, a curvilinear grid is defined and the variable values at grid points adjacent to a boundary are forced or interpolated to satisfy the boundary conditions. The MPM is used for solving the equations of solid structure and communicates with the fluid through appropriate interface‐boundary conditions. The governing flow equations are discretized on a non‐staggered grid layout using second‐order accurate finite‐difference formulas. The discrete equations are integrated in time via a second‐order accurate dual time stepping, artificial compressibility scheme. Unstructured, triangular meshes are employed to discretize the complex surface of the IBs. The nodes of the surface mesh constitute a set of Lagrangian control points used for tracking the motion of the flexible body. The equations of the solid body are integrated in time via the MPM. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at stationary curvilinear grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The influence of the fluid on the body is defined through pressure and shear stresses acting on the surface of the body. The HIBM & MPM approach is validated for FSI problems by solving for a falling rigid and flexible sphere in a fluid‐filled channel. The behavior of a capsule in a shear flow was also examined. Agreement with the published results is excellent. Copyright © 2007 John Wiley & Sons, Ltd.     

Mathematical modelling of two‐phase non‐Newtonian flow in a helical pipe

Governing equations for a two‐phase 3D helical pipe flow of a non‐Newtonian fluid with large particles are derived in an orthogonal helical coordinate system. The Lagrangian approach is utilized to model solid particle trajectories. The interaction between solid particles and the fluid that carries them is accounted for by a source term in the momentum equation for the fluid. The force‐coupling method (FCM), developed by M.R. Maxey and his group, is adopted; in this method the momentum source term is no longer a Dirac delta function but is spread on a numerical mesh by using a finite‐sized envelop with a spherical Gaussian distribution. The influence of inter‐particle and particle–wall collisions is also taken into account. Copyright © 2005 John Wiley & Sons, Ltd.     

Validation of a fluid–structure interaction model for a bileaflet mechanical heart valve

A fluid-structure interaction (FSI) model for heart valve simulation is presented. In a partitioned framework, separate fluid and structure solvers are weakly coupled, which in combination with the use of artificial compressibility in the fluid solver, leads to a stable and efficient approach. An Arbitrary Lagrangian Eulerian formulation is employed in the fluid solver to permit the accurate calculation of shear stresses next to the valve boundary. The mesh quality is maintained through a combination of smoothing and local remeshing in 3D. The FSI algorithm is validated on experiments of an idealised quasi-2D mechanical heart valve, and the efficiency of the remeshing approach is demonstrated on a realistic 3D heart-valve geometry.     

Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

Two different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.     

A coupled finite volume method for the solution of flow processes on complex geometries

CFD modelling of ‘real‐life’ thermo‐fluid processes often requires solutions in complex three‐dimensional geometries, which can result in meshes containing aspects that are badly distorted. Cell‐centred finite volume methods (CC‐FV), typical of most commercial CFD tools, are computationally efficient, but can lead to convergence problems on meshes that feature cells with highly non‐orthogonal shapes. The control volume‐finite element method (CVFE) uses a vertex‐based approach and handles distorted meshes with relative ease, but is computationally expensive. A combined vertex‐based—cell‐centre technique (CFVM), detailed in this paper, allows solutions on distorted meshes where purely cell‐centred solutions procedures fail. The method utilizes the ability of the vertex‐based approach to resolve the flow field on a distorted mesh, enabling well established cell‐centred physical models to be employed in the solution of other transported quantities. The vertex‐based flow code is verified against a manufactured 3D solution and error norms are compared on meshes with various degrees of distortion. The CFVM method is validated with benchmark solutions for thermally driven flow and turbulent flow. Finally, the method is illustrated on three‐dimensional turbulent flow over an aircraft wing on a distorted mesh where purely cell‐centred techniques fail. The CFVM is relatively straightforward to embed within generic CC based CFD tools allowing it to be employed in a wide variety of processing applications. Copyright © 2006 John Wiley & Sons, Ltd.     

A time-dependent numerical analysis of flow in a mechanical heart valve: Comparison with experimental results

There is a great need to fabricate heart valves that have similar haemodynamic properties with the natural ones. Towards this goal, we examine the dynamics of fluid flow in a mechanical heart valve with one leaflet. The fluid is incompressible and Newtonian and the leaflet is a neo-Hookean material. The Arbitrary Lagrangian Eulerian method is used to model the fluid-leaflet interaction, and the system of equations is solved using the Finite Element method. The pseudo solid approach along with a set of algebraic equations are used to deform the mesh, while care is taken to avoid remeshing of the domain, at the moment of valve closure. The computational results are compared against the experimental results, and we find an excellent agreement for the time period of valve closure, the time the valve is fully opened, and the value of the maximum valve opening angle. This study indicates that the present model is capable of describing the valve dynamics in physiological geometries.     

An immersed boundary method for unstructured meshes in depth averaged shallow water models

The representation of geometries as buildings, flood barriers or dikes in free surface flow models implies tedious and time‐consuming operations in order to define accurately the shape of these objects when using a body fitted numerical mesh. The immersed boundary method is an alternative way to define solid bodies inside the computational domain without the need of fitting the mesh boundaries to the shape of the object. In the direct forcing immersed boundary method, a solid body is represented by a grid of Lagrangian markers, which define its shape and which are independent from the fluid Eulerian mesh. This paper presents a new implementation of the immersed boundary method in an unstructured finite volume solver for the 2D shallow water equations. Moving least‐squares is used to transmit information between the grid of Lagrangian markers and the fluid Eulerian mesh. The performance of the proposed implementation is analysed in three test cases involving different flow conditions: the flow around a spur dike, a dam break flow with an isolated obstacle and the flow around an array of obstacles. A very good agreement between the classic body fitted approach and the immersed boundary method was found. The differences between the results obtained with both methods are less relevant than the errors because of the intrinsic shallow water assumptions. Copyright © 2015 John Wiley & Sons, Ltd.     

Mono‐block and non‐matching multi‐block structured mesh adaptation based on aerodynamic functional total derivatives for RANS flow

An enhanced goal‐oriented mesh adaptation method is presented based on aerodynamic functional total derivatives with respect to mesh nodes in a Reynolds‐Averaged Navier‐Stokes (RANS) finite‐volume mono‐block and non‐matching multi‐block‐structured grid framework. This method falls under the category of methods involving the adjoint vector of the function of interest. The contribution of a Spalart–Allmaras turbulence model is taken into account through its linearization. Meshes are adapted accordingly to the proposed indicator. Applications to 2D RANS flow about a RAE2822 airfoil in transonic, and detached subsonic conditions are presented for the drag coefficient estimation. The asset of the proposed method is patent. The obtained 2D anisotropic mono‐block mesh well captures flow features as well as global aerodynamic functionals. Interestingly, the constraints imposed by structured grids may be relaxed by the use of non‐matching multi‐block approach that limits the outward propagation of local mesh refinement through all of the computational domain. The proposed method also leads to accurate results for these multi‐block meshes but at a fraction of the cost. Finally, the method is also successfully applied to a more complex geometry, namely, a mono‐block mesh in a 3D RANS transonic flow about an M6 wing. Copyright © 2016 John Wiley & Sons, Ltd.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulation of flow around a thin,flexible obstruction by means of a deforming grid overlapping a fixed grid

This paper presents a numerical method for simulation of coupled flows, in which the fluid interacts with a thin deformable solid, such as flows in cardiovascular valves. The proposed method employs an arbitrary Lagrangian–Eulerian (ALE) method for flow near the solid, embodied in the outflow domain in which the mesh is fixed. The method was tested by modelling a two‐dimensional channel flow with a neo‐Hookean obstacle, an idealization of the coupled flow near a cardiovascular valve. The effects of the Reynolds number and the dimensionless elastic modulus of the material on the wall shear stress, the size of the downstream reverse flows, and the velocity and pressure profiles were investigated. The deformation of the obstacle, the pressure drop across the obstacle, and the size of the top reverse flow increased as the Reynolds number increased. Conversely, increasing the elastic modulus of the obstacle decreased the deformation of the obstacle and the size of the top reverse flows, but did not affect the pressure drop across the obstacle over the range studied. Copyright © 2007 John Wiley & Sons, Ltd.     

A new mesh relaxation approach and automatic time‐step control method for boundary integral simulations of a viscous drop

We present a numerical methodology for the simulation of a viscous drop under simple shear flows by using the boundary integral method. The present work treats only a single drop in an unbounded fluid‐flow, but the results can be directly applied to studies on the rheology of dilute emulsions, in which the hydrodynamic interactions between two or more drops can be neglected. Singular and non‐singular integral representations of the velocity field are considered. Several aspects of the method are presented, including a new mesh relaxation approach and an automatic time‐step control method. The relaxation strategy is used in order to contain the distortion of the mesh and is performed by using relaxation iterations in a virtual temporal march between each physical time step of the simulation and monitoring the standard deviation of the areas of the elements. The automatic time‐step control method uses a global quantity related to the drop deformation in order to automatically set the temporal integration time step. It is carried out in a way to keep the local integration error less than a given tolerance. This strategy reduces the computational cost of the simulation by dramatically reducing the number of time steps in the temporal integration process. Copyright © 2016 John Wiley & Sons, Ltd.     

A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion

Computational fluid mechanics techniques for examining free surface problems in two‐dimensional form are now well established. Extending these methods to three dimensions requires a reconsideration of some of the difficult issues from two‐dimensional problems as well as developing new formulations to handle added geometric complexity. This paper presents a new finite element formulation for handling three‐dimensional free surface problems with a boundary‐fitted mesh and full Newton iteration, which solves for velocity, pressure, and mesh variables simultaneously. A boundary‐fitted, pseudo‐solid approach is used for moving the mesh, which treats the interior of the mesh as a fictitious elastic solid that deforms in response to boundary motion. To minimize mesh distortion near free boundary under large deformations, the mesh motion equations are rotated into normal and tangential components prior to applying boundary conditions. The Navier–Stokes equations are discretized using a Galerkin–least square/pressure stabilization formulation, which provides good convergence properties with iterative solvers. The result is a method that can track large deformations and rotations of free surface boundaries in three dimensions. The method is applied to two sample problems: solid body rotation of a fluid and extrusion from a nozzle with a rectangular cross‐section. The extrusion example exhibits a variety of free surface shapes that arise from changing processing conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

Adaptive Lagrangian–Eulerian computation of propagation and rupture of a liquid plug in a tube

Liquid plug propagation and rupture occurring in lung airways can have a detrimental effect on epithelial cells. In this study, a numerical simulation of a liquid plug in an infinite tube is conducted using an Eulerian–Lagrangian approach and the continuous interface method. A reconstruction scheme is developed to allow topological changes during plug rupture by altering the connectivity information about the interface mesh. Results prior to the rupture are in reasonable agreement with the study of Fujioka et al. in which a Lagrangian method is used. For unity non‐dimensional pressure drop and a Laplace number of 1000, rupture time is shown to be delayed as the initial precursor film thickness increases and rupture is not expected for thicknesses larger than 0.10 of tube radius. During the plug rupture process, a sudden increase of mechanical stresses on the tube wall is recorded, which can cause tissue damage. The peak values of those stresses increase as the initial precursor film thickness is reduced. After rupture, the peaks in mechanical stresses decrease in magnitude as the plug vanishes and the flow approaches a fully developed behavior. Increasing initial pressure drop is shown to linearly increase maximum variations in wall pressure and shear stress. Decreasing the pressure drop and increasing the Laplace number appear to delay rupture because it takes longer for a fluid with large inertial forces to respond to the small pressure drop. Copyright © 2010 John Wiley & Sons, Ltd.     

An ALE mesh movement scheme for long‐term in‐flight ice accretion

The rather irregular shapes that glaze ice may grow into while accreting over the surface of an aircraft represent a major difficulty in the numerical simulation of long periods of in‐flight icing. There is a constant need for remeshing: a wasteful procedure. In the framework of ALE formulations, a mesh movement scheme is presented, in which frame and elasticity analogies are loosely coupled. The resulting deformed mesh preserves the quality of elements, especially in the near‐wall region, where accurate prediction of heat flux and shear stresses is required. The proposed scheme handles mesh deformation in a computationally efficient manner by localizing the mesh deformation. The 2D problem of ice accretion over single and multi‐element airfoils is considered here as a numerical experiment. Experimentally measured glaze ice shapes were used to evaluate the performance of the present approach. Copyright © 2011 John Wiley & Sons, Ltd.     

Anisotropic goal‐oriented error analysis for a third‐order accurate CENO Euler discretization

In this paper, a central essentially non‐oscillatory approximation based on a quadratic polynomial reconstruction is considered for solving the unsteady 2D Euler equations. The scheme is third‐order accurate on irregular unstructured meshes. The paper concentrates on a method for a metric‐based goal‐oriented mesh adaptation. For this purpose, an a priori error analysis for this central essentially non‐oscillatory scheme is proposed. It allows us to get an estimate depending on the polynomial reconstruction error. As a third‐order error is not naturally expressed in terms of a metric, we propose a least‐square method to approach a third‐order error by a quadratic term. Then an optimization problem for the best mesh metric is obtained and analytically solved. The resulting mesh optimality system is discretized and solved using a global unsteady fixed‐point algorithm. The method is applied to an acoustic propagation benchmark.     

改进虚拟边界算法在超声速流动问题求解中的应用

提出了一种改进的虚拟单元浸没边界法, 并与一种高阶格式的有限差分算法相结合, 运用于求解超声速复杂几何绕流问题.该算法的核心思想在于在固体边界的内部和外部分别施加满足边界关系的作用点, 使得几何边界离散更加细化, 起到了壁面附近网格局部加密的作用.采用源空间内流体点作为反距离插值算法的重构点, 有效避免了插值点数目过少而与作用点相重合情况.通过对二维激波反射现象 (马赫数为 2.81) 和三维超声速球体绕流问题 (马赫数为 1.2) 的数值模拟, 与实验结果对比表明, 本文改进算法相对一般的虚拟边界法来说能显著提高数值精度, 减小计算误差.计算结果揭示了球体绕流中剪切层、压缩波系和尾迹的相互作用导致自由剪切层失稳的机理.剪切层厚度和湍流雷诺脉动经历了线性增长、大幅度震荡和小幅度波动三个阶段, 导致剪切层表面褶皱因子变化呈指数规律增长.其湍流结构表现出明显的各向异性, 具体在流向雷诺正应力在湍流脉动中占主导地位, 激波的压缩作用对不同方向雷诺正应力的影响存在空间迟滞效应.